What is your favorite example of a celebrated mathematical fact that had a hard time to become accepted by the community, but after overcoming some initial "resistance" quickly took on?
It can be a theorem, a proof method, an algorithm or a definition, that is
- widely known and
- very useful in the present day,
- less than 99 years old; this is in order to avoid examples from the very distant past, such the difficulties Grassmann's work had being accepted
but which at the time of its inception was not appreciated, misunderstood or ignored by the mathematical community, before it became mainstream or inspired other research which in turn became mainstream. This is a partial converse question to this one that asks for mathematical facts that were quickly accepted but then discarded by the community. This and this question are somewhat related, but former focuses on people (resp. their entire works, see Grassmann) not being accepted, rather then individual results, whereas the latter solely on famous articles rejected by journal; also, the results that are being mentioned in these links are often rather old and do not fit this question.
Example. Numerical optimization: The first quasi-Newton algorithm was discovered in 1959 and "was not accepted for publication; it
remained as a technical report for more than thirty years until it appeared in the first issue of the SIAM Journal on Optimization in 1991" (Nocedal & Wright, Numerical Optimization).
But the algorithm inspired a slew of other variants, has been cited over 2000 times to this day and quasi-Newton type algorithms are still state-of-the-art in for certain optimization problems.
Best Answer
The Selberg integral was proved in a 1944 paper of Selberg, after being stated without proof in a 1941 paper. The paper was in Norwegian, and was also in a journal that would have been of little interest to the research community:
This result was little-used, being used in one paper in 1953.
A closely related integral then appeared in random matrix theory. Mehta and Dyson gave a conjectural value for this integral, publicizing this conjecture as an open problem in a paper in 1963, a textbook in 1967, and the SIAM Review in 1974. However, no one remembered Selberg's work and thought to apply it.
Finally in 1976 Bombieri came across another similar integral when studying a different topic (prime numbers). He went to discuss his overall work on the distribution of prime numbers with Selberg, because of Selberg's expertise in number theory, and Selberg then mentioned his integral, which Bombieri used to solve his problem.
This was after Bombieri was informed by Spencer about the relationship of his integral to a third topic (the Coulomb gas), motivating him to ask Dyson about it, at which point Dyson explained the connection to random matrices, and thus Bombieri was able to prove the conjecture in random matrix theory as well.
Since then, the result has found further use and development, and is now widely-known.
My source for all these details is the paper The importance of the Selberg integral by Peter J. Forrester and S. Ole Warnaar